p˙m=∑n(wmnpn−wnmpm)
wnmwmn=pn⋆pm⋆=exp(−Tϵm−ϵn)
S˙≥TQ˙
S(P)=f(∑mg(pm))
Maximize S(p) subject to constraint that p is normalized and expected energy has a given value
Solution: MaxEnt distribution: pm⋆=(g′)−1(Cfα+βϵm), Cf=f′(∑mg(pm))
U=∑mpmϵm
S=f(∑mg(pm))
S=−∑mpmlogpm
p˙m=∑n[J(wmn,pn)−J(wnm,pm)]
p˙m=∑n(wmnpn−wnmpm)
∑mp˙m=0
wmn
[J(wmn,pn)−J(wnm,pm)]
pm⋆=(g′)−1(Cfα+βϵm)
pm⋆=exp(−α−βϵm)
J(wmn,pn⋆)=J(wnm,pm⋆)
wmnpn⋆=wnmpm⋆
dtdS=S˙i+S˙e
S˙i≥0 and S˙i=0⇔J(wmn,pn)=J(wnm,pm) ∀ m,n
S˙e=T1∑mp˙mϵm=TQ˙
J(wmn,pn)=ψ(j(wmn)−g′(pn))
j - arbitrary function
ψ - increasing function
p˙m=∑n(wmnpn−wnmpm)
Jmn=wmnpn=exp(logwmn+logpn)
⇒g′(pn)=−log(pn)
⇒S=−∑npnlogpn
H=Hsystem+Hbath
λHbath(x1,…,xn)=Hbath(λ1/a1x1,…,λ1/anxn)
p(E) ∝∫δ(E−Hbath)dx1…dxn
p(E) ∝(1−(q−1)βE)1/(q−1)
S=1−q1(∑mpmq−pm)
⇒g′(pm)=1−qqpmq−1−1
Jmn=ψ(j(wmn)+q−1qpmq−1−1)
S˙=−∑mp˙mlogpm
=−21∑mn(wmnpn−wnmpm)logpnpm
= S˙i21mn∑(wmnpn−wnmpm)logwnmpmwmnpn
+S˙e21mn∑(wmnpn−wnmpm)logwnmwmn
= S˙i+S˙e≥TQ˙
S˙=Cf∑mp˙mg′(pm)
=2Cf∑mn(Jmn−Jnm)(g′(pm)−g′(pn))
=2Cf∑mn(Jmn−Jnm)(Φmn−Φnm)
+2Cf∑mn (Jmn−Jnm)(g′(pm)+Φnm−g′(pn)−Φmn)
=S˙i2Cfmn∑(Jmn−Jnm)(ϕ(Jmn)−ϕ(Jnm))
+S˙e2Cfmn∑ (Jmn−Jnm)(g′(pm)+ϕ(Jnm)−g′(pn)−ϕ(Jmn))
S˙i⇒ϕ− increasing
S˙e⇒Cf[g′(pm)+ϕ(Jnm)−g′(pn)−ϕ(Jmn)]=Tϵn−ϵm
⇒ϕ(Jmn)=j(wmn)−g′(pn)
⇒Jmn=ψ(j(wmn)−g′(pn)), ψ=ϕ−1 - increasing □.
j(wmn)−j(wnm)=CfTϵn−ϵm
β=T1